# Prediction scores

#### Finn Lindgren

#### Corrections and code updates 2025-04-09

    library(dplyr)
    #>
    #> Attaching package: 'dplyr'
    #> The following objects are masked from 'package:stats':
    #>
    #>     filter, lag
    #> The following objects are masked from 'package:base':
    #>
    #>     intersect, setdiff, setequal, union
    library(scoringRules)

## Proper posterior prediction scores

A prediction score \(S(F,y)\) evaluates some measure of closeness between a prediction distribution identified by \(F\), and an observed value \(y\).

### A basic score, and motivating remarks

A common score is the Squared Error, \[ S_\text{SE}(F,y) = [y - \mathbb{E}_F(Y)]^2, \] and we would like predictions to have low values for \(S_\text{SE}(F,y)\), indicating a “good” prediction, in that specific sense that puts a penalty on the squared deviation of the prediction mean from the true observed value. We can imagine constructing other such scoring functions that penalise other aspects of the prediction.

A score where “lower is better” is called negatively oriented, and a score where “higher is better” is called positively oriented. One can always turn one type of score into the other by changing the sign, so to simplify the presentation, we’ll make all scores be negatively oriented, like the squared error.

We often care about the prediction uncertainty and not just the mean (or at least we should care!). Just adding a prediction variance penalty to the squared error wouldn’t be useful, as we could then construct a new, “better”, prediction by reducing the stated prediction variance to zero. This would understate the real prediction uncertainty, so wouldn’t be a fair scoring approach for comparing different prediction models. In the next section, we make this fairness idea more precise.

### Proper and strictly proper scores

The expected value under a distribution identified by \(G\) is denoted \(S(F,G):=\mathbb{E}_{Y\sim F}[S(F,Y)]\). For a negatively oriented score, we seek scoring functions that are fair, in the sense that one cannot, on average, make a better prediction that that which generated the data. This requires \(S(F,G)\geq S(G,G)\) for all predictive distributions \(F\) and any distribution \(G\). Such scores are call proper. If in addition, equality of the score expectations only hold when \(F=G\), the score is strictly proper.

Non-strict proper scores ignore some aspect of the prediction, typically by only being sensitive to some summary information, such as the mean, median, and/or variance.

It’s notable that proper scores retain their properness under affine transformations, with just potential changes in whether they are positively or negatively oriented. If \(S(F,y)\) is a proper score, then \[ S'(F,y) = a + b S(F,y),\quad a,b\in\mathbb{R}, \] is also a proper score, with the same orientation if \(b>0\) and the opposite orientation if \(b<0\). The degenerate case \(b=0\) gives the score \(a\) to all predictions, which is technically a proper score (you cannot to better than an ideal prediction), but a useless one (an ideal prediction is no better than any other prediction).

### Examples

- log-score: \(S_\text{log}(F,y) = -\log\{p_F(y)\}\), where \(p_F(y)\) is a predictive pdf or pmf for \(y\), is a strictly proper score.

- Squared Error: \(S_\text{SE}(F,y) = [y - \mathbb{E}_F(Y)]^2\) is a proper score

- Brier score:

  - Binary events: \[S_\text{Brier}(F,z) = [z - \mathbb{P}_F(Z = 1)]^2,\] where \(Z\in\{0,1\}\) is a binary event indicator, is a strictly proper score for the event prediction, but non-strict with respect to any underlying outcome \(y\) generating the event indicator, e.g. via \(z=I(y=0)\).
  - Class indicator events: if \(y\in\{1,\dots,K\}\) is a class category outcome, the Brier score can be generalised to \[S_\text{MultiBrier}(F,y) = \sum_{k=1}^K [I(y = k) - \mathbb{P}_F(Y=k)]^2\] which can be seen as the Squared Error for the Multinomial prediction model for \(z_k=I(y=k)\), with \(\{z_1,\dots,z_K\}\sim\text{Multinomial}(1,\{p_1,\dots,p_K\})\), where \(p_k=\mathbb{P}_F(Y=k)\). Sometimes, the sum is normalised by \(1/K\). This generalised Brier score is a proper score.

- Dawid-Sebastiani: \[S_\text{DS}(F,y)=[y - \mathbb{E}_F(Y)]^2 / \mathbb{V}_F(Y) + \log[\mathbb{V}_F(Y)]\] is a proper score. It’s derived from the strictly proper log-score of a Gaussian prediction, but it’s also a non-strict proper score for other distributions. It has the advantage that it only involves the predictive mean and variance, making it computable also in cases when log-densities are hard to obtain. Since it’s based on the symmetric Gaussian distribution, it tends to be affected by skewness, so should be applied with care in such cases.

- Absolute (Median) Error: \(S_\text{AE}(F,y)=|y - \text{median}_F|\) is a proper score, with expectation minimised when the medians of \(F\) and \(G\) match. Note that \(|y-\mathbb{E}_F(Y)|\), the absolute error with respect to the expectation, is not a proper score! Another way of expressing this is that if \(|y-m_F|\) is a proper score with respect to the median, i.e. it is proper when \(m_F\) is taken to be the median of \(F\), and not some other point prediction. Unfortunately, the predictive median is not readily available, and typically requires sampling from the observation level predictive distribution, or estimating the median of the a mixture distribution. In the applied literature, this distinction is often overlooked, and the predictive mean is inserted into both the SE and AE scores, making the resulting AE score comparisons less clear than they could be.

- CRPS (Continuous Ranked Probability Score):

  \[ S_\text{CRPS}(F,y)= \int_{-\infty}^\infty [\mathbb{P}_F(Y \leq x) - I(y \leq x)]^2 \,\mathrm{d}x \] This is a strictly proper score, related to the absolute error of point predictions.

Other scores include the Interval score that is minimised for short prediction intervals with the intended coverage probability, and the Quantile score, that generalises the Absolute Median Error to other quantiles than the median.

### Improper scores

We’ve seen that some scores are strictly proper, and others are only proper scores, sensitive to specific aspects of the predictive distribution, such as mean, median, and/or variance.

In contrast, improper scores do not fulfil the fairness idea. Such scores include the the aforementioned penalised squared error, \([y-\mathbb{E}(Y)]^2+\mathbb{V}_F(Y)\), but also the probability/density function, \(p_F(y)\). The latter might come as a surprise, as the log-score is proper.

### Mean error/score

Up to this point, we only considered individual scores. When summarising predictions \(\{F_i\}\) for a collection of observations \(\{y_i\}\), we usually compute the mean score, \[ S(\{F_i\},\{y_i\}) = \frac{1}{N}\sum_{i=1}^N S(F_i,y_i) . \]

When comparing two different prediction models \(F\) and \(F'\), the scores are dependent with respect to the observations \(y_i\). This means that in order to more easily handle the score variability in the comparison, we should treat it as a paired sample problem. The pairwise score differences are given by \[ S(F_i,F'_i,y_i) = S(F_i,y_i) - S(F'_i,y_i) . \] It’s also much more reasonable to make conditional independence assumptions about these differences, than for the plain score values \(S(F_i,y_i)\); \[ \mathbb{V}_{\{y_i\}\sim G}\left[\frac{1}{N}\sum_{i=1}^N S(F_i,y_i)\right] = \frac{1}{N^2}\sum_{i=1}^N\sum_{j=1}^N\mathbb{C}_{\{y_i\}\sim G}\left[S(F_i,y_i),S(F_j,y_j)\right] \] but \[ \mathbb{V}_{\{y_i\}\sim G}\left[\frac{1}{N}\sum_{i=1}^N S(F_i,y_i) - S(F'_i,y_i)\right] \approx \frac{1}{N^2}\sum_{i=1}^N\mathbb{V}_{y_i\sim G_i}\left[ S(F_i,y_i) - S(F'_i,y_i)\right]. \]

Note that taking the average of prediction scores, or averages of prediction score differences, is quite different from assessing summary statistics of the collection of predictions, since the scores are individual for each observation; we’re not assessing the collective value distribution, as that might be misleading. For example, consider a spatial model where he estimated procession has an empirical distribution of the predictive means that matches that of the observed data. Scores based on the marginal empirical distribution would not be able to detect if the location of the values is maximally different to the actual locations, whereas averages of individual scores would be sensitive to this.

## Poisson model example

Consider a model with Poisson outcomes \(y\), conditionally on a log-linear predictor \(\lambda=\exp(\eta)\), where \(\eta\) is some linear expression in latent variables.

The posterior predictive distributions are Poisson mixture distributions across the posterior distribution of \(\lambda\), \(p(\lambda|\text{data})\).

For illustration purposes, consider a simple Poisson model with a single covariate \(x\) and a linear predictor \(\eta = \beta_0 + \beta_1 x\).

    df_pois <- tibble::tibble(
      x = rnorm(50),
      y = rpois(length(x), exp(2 + 1 * x))
    )
    fit_pois <- bru(
      y ~ Intercept(1) + x,
      family = "poisson",
      data = df_pois
    )
    newdata_pois <- tibble::tibble(
      x = rnorm(100),
      y = rpois(100, exp(2 + 1 * x))
    )

### Moment scores

For the Squared Error and Dawid-Sebastiani scores, we’ll need the posterior expectation and variance: \[ \mathbb{E}(Y|\text{data}) = \mathbb{E}[\mathbb{E}(Y|\lambda,\text{data}) | \text{data}] = \mathbb{E}(\lambda | \text{data}) \] and \[ \mathbb{V}(Y|\text{data}) = \mathbb{E}[\mathbb{V}(Y|\lambda,\text{data}) | \text{data}] + \mathbb{V}[\mathbb{E}(Y | \lambda, \text{data}) | \text{data}] = \mathbb{E}[\lambda | \text{data}] + \mathbb{V}[\lambda | \text{data}] \] i.e. the sum of the posterior mean and variance for lambda.

The SE and DS scores are therefore relatively easy to compute after estimating a model with `inlabru`. You just need to estimate the posterior mean and variance with `predict()` for each test data point. If `Intercept+x` is the expression for the linear predictor, and `newdata` holds the covariate information for the prediction points, run

    pred_pois <- predict(
      fit_pois,
      newdata_pois,
      formula = ~ exp(Intercept + x),
      n.samples = 2000
    )
    post_E <- pred_pois$mean
    post_Var <- pred_pois$mean + pred_pois$sd^2
    scores <- data.frame(
      SE = (newdata_pois$y - post_E)^2,
      DS = (newdata_pois$y - post_E)^2 / post_Var + log(post_Var)
    )
    summary(scores)
    #>        SE                  DS
    #>  Min.   :  0.00022   Min.   : 0.01483
    #>  1st Qu.:  0.55470   1st Qu.: 1.95030
    #>  Median :  3.20672   Median : 2.78956
    #>  Mean   : 18.17626   Mean   : 3.21006
    #>  3rd Qu.: 11.47258   3rd Qu.: 3.69239
    #>  Max.   :560.38827   Max.   :10.58477

### Log-Probability and log-density scores

The full log-score can actually also be estimated/computed in a similar way. We seek, for a fixed observation y, \(\log[\mathbb{P}(Y = y | \text{data})]\) The probability is \[ \mathbb{P}(Y = y | \text{data}) = \mathbb{E}[\mathbb{P}(Y = y | \lambda, \text{data}) | \text{data}], \] so we can estimate it using `predict()`:

    pred_pois <- predict(fit_pois,
      newdata_pois,
      formula = ~ dpois(y, lambda = exp(Intercept + x)),
      n.samples = 2000
    )
    log_score <- -log(pred_pois$mean)
    summary(log_score)
    #>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
    #>  0.5647  1.8032  2.2839  2.4738  2.7616  6.0177

to estimate the log_score (increase `n.samples` if needed for sufficiently small Monte Carlo error). Approximate Monte Carlo confidence intervals for the scores can be obtained via the Monte Carlo standard errors:

    head(
      data.frame(
        log_S = log_score,
        lower = -log(pred_pois$mean + 2 * pred_pois$mean.mc_std_err),
        upper = -log(pred_pois$mean - 2 * pred_pois$mean.mc_std_err)
      )
    )
    #>      log_S    lower    upper
    #> 1 2.845689 2.832964 2.858577
    #> 2 2.173229 2.168151 2.178332
    #> 3 2.172802 2.165996 2.179656
    #> 4 2.757201 2.752975 2.761445
    #> 5 2.412056 2.410966 2.413148
    #> 6 2.265886 2.264522 2.267252

### CRPS

Yet another option would be to use the CRPS, which for each prediction value \(y_i\) would be \[ S_\text{CRPS}(F_i,y_i) = \sum_{k=0}^\infty [\mathbb{P}(Y_i \leq k |\text{data}) - I(y_i \leq k)]^2 \] For this, one would first need to get \(\mathbb{P}(Y \leq k | \text{data})\) from a predict call with `ppois(k, lambda = exp(eta))`, for a vector \(k=0,1\dots,K\), for each \(y_i\), for some sufficiently large \(K>\max_i{y_i}\) for the remainder to be negligible. However, to avoid repeated `predict()` calls for each \(y_i\), the storage requirements is of order \((K+1) \times N\times N_\text{samples}\). To avoid that, one option would be to reformulate the estimator into a recursive estimator, so that batches of simulations could be used to iteratively compute the estimator.

A basic estimator can proceed as follows:

Define \(K\geq K_0=\max_i(y_i)\) sufficiently large for the posterior predictive probability above \(K\) to be negligible. Perhaps a value like \(K=K_0+4\sqrt{K_0}\) might be sufficient. You can check afterwards, and change if needed.

1. Simulate samples from \(\lambda^{(j)}\sim p(\lambda|\text{data})\) using `generate()` (size \(N\times N_\text{samples}\)).
2. For each \(i=1,\dots,N\), use the samples to estimate the residuals \(r_{ik}=\mathbb{P}(Y\leq k|\text{data})-I(y_i\leq k)\), for \(k\in 0,1,2,\dots,K\), with

\[ \widehat{r}_{ik} = \frac{1}{N_\text{samples}} \sum_{j=1}^{N_\text{samples}} \{ \mathbb{P}(Y\leq k|\lambda^{(j)}_i) - I(y_i\leq k) \} . \] 3. Compute

\[ \widehat{S}_\text{CRPS}(F_i,y_i) = \sum_{k=0}^{K} \widehat{r}_{ik}^2 \]

    # some large value, so that 1-F(K) is small
    max_K <- ceiling(max(newdata_pois$y) + 4 * sqrt(max(newdata_pois$y)))
    k <- seq(0, max_K)
    kk <- rep(k, times = length(newdata_pois$y))
    i <- seq_along(newdata_pois$y)
    pred_pois <- generate(fit_pois, newdata_pois,
      formula = ~ {
        lambda <- exp(Intercept + x)
        ppois(kk, lambda = rep(lambda, each = length(k)))
      },
      n.samples = 2000
    )
    results <- data.frame(
      i = rep(i, each = length(k)),
      k = kk,
      Fpred = rowMeans(pred_pois),
      residuals =
        rowMeans(pred_pois) - (rep(newdata_pois$y, each = length(k)) <= kk)
    )
    # Check that the cutoff point K has nearly probability mass 1 below it,
    # for all i:
    min(results |> dplyr::filter(k == max_K) |> pull(Fpred))
    #> [1] 1

    crps_scores <-
      (results |>
        group_by(i) |>
        summarise(crps = sum(residuals^2), .groups = "drop") |>
        pull(crps))
    summary(crps_scores)
    #>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
    #>  0.1967  0.7630  1.1877  1.9699  2.0723 18.2134

## Normal model example

Consider a model with Normal outcomes \(y\sim N(\eta,\sigma^2_y)\), conditionally on a linear predictor \(\lambda=\eta\), where \(\eta\) is some linear expression in latent variables.

The posterior predictive distributions are Normal mixture distributions across the posterior distribution of \(\eta\) and \(\sigma^2_y\), \(p(\lambda,\sigma^2_y|\text{data})\).

For illustration purposes, consider a simple Poisson model with a single covariate \(x\) and a linear predictor \(\eta = \beta_0 + \beta_1 x\).

    true_sigma_y <- 0.1
    df_normal <- tibble::tibble(
      x = rnorm(50),
      y = rnorm(length(x), 2 + 1 * x, sd = true_sigma_y)
    )
    fit_normal <- bru(
      y ~ Intercept(1) + x,
      family = "normal",
      data = df_normal
    )
    newdata_normal <- tibble::tibble(
      x = rnorm(100),
      y = rnorm(100, 2 + 1 * x, sd = true_sigma_y)
    )

### Moment scores and log-score

For the Squared Error and Dawid-Sebastiani scores, we’ll need the posterior expectation and variance: \[ \mathbb{E}(Y|\text{data}) = \mathbb{E}[\mathbb{E}(Y|\eta, \sigma_y^2,\text{data}) | \text{data}] = \mathbb{E}(\eta | \text{data}) \] and \[ \mathbb{V}(Y|\text{data}) = \mathbb{E}[\mathbb{V}(Y|\eta,\sigma_y^2,\text{data}) | \text{data}] + \mathbb{V}[\mathbb{E}(Y | \eta,\sigma_y^2,\text{data}) | \text{data}] = \mathbb{E}[\sigma_y^2 | \text{data}] + \mathbb{V}[\eta | \text{data}] \] i.e. the sum of the posterior mean of \(\sigma_y^2\) and the variance of \(\eta\).

The SE and DS scores are therefore relatively easy to compute after estimating a model with `inlabru`, as well as the log-score. You just need to estimate the posterior mean and variance with `predict()` for each test data point. If `Intercept+x` is the expression for the linear predictor, and `newdata` holds the covariate information for the prediction points, run

    pred_normal <- predict(fit_normal, newdata_normal,
      formula = ~ {
        E <- Intercept + x
        V <- 1 / Precision_for_the_Gaussian_observations
        list(
          eta = E,
          sigma_y_2 = V,
          dens = dnorm(y, mean = E, sd = sqrt(V))
        )
      },
      n.samples = 2000
    )
    post_E <- pred_normal$eta$mean
    post_Var <- pred_normal$sigma_y_2$mean + pred_normal$eta$sd^2
    scores_normal <- data.frame(
      SE = (newdata_normal$y - post_E)^2,
      DS = (newdata_normal$y - post_E)^2 / post_Var + log(post_Var),
      LS = -log(pred_normal$dens$mean)
    )
    summary(scores_normal)
    #>        SE                  DS               LS
    #>  Min.   :1.076e-06   Min.   :-4.399   Min.   :-1.2958
    #>  1st Qu.:6.345e-04   1st Qu.:-4.315   1st Qu.:-1.2543
    #>  Median :3.604e-03   Median :-4.101   Median :-1.1382
    #>  Mean   :9.792e-03   Mean   :-3.598   Mean   :-0.8810
    #>  3rd Qu.:1.299e-02   3rd Qu.:-3.343   3rd Qu.:-0.7407
    #>  Max.   :6.887e-02   Max.   : 0.985   Max.   : 1.4173

Here, we’ve again used the negated log-score, so that small values are “better” for all three scores. Since the DS score is based on the log-score for Normal predictions, the conditional DS and LS scores differ only by a constant offset and scaling. However, since the full posterior predictions are Normal mixtures, the DS and LS scores for the full posterior predictive distribution are not equivalent.

As for the Poisson case, we can estimate the log-score uncertainty:

    head(
      data.frame(
        log_S = scores_normal$LS,
        lower = -log(pred_normal$dens$mean + 2 * pred_normal$dens$mean.mc_std_err),
        upper = -log(pred_normal$dens$mean - 2 * pred_normal$dens$mean.mc_std_err)
      )
    )
    #>         log_S       lower        upper
    #> 1 -0.01882446 -0.03111731 -0.006378606
    #> 2 -0.64190251 -0.64936553 -0.634383383
    #> 3 -1.27824731 -1.28299230 -1.273479697
    #> 4 -1.05447506 -1.06204747 -1.046844879
    #> 5 -0.63967299 -0.65176544 -0.627432518
    #> 6 -1.23794301 -1.24273406 -1.233128894

## Posterior expectation of conditional scores

In some cases, one might be tempted to consider posterior distribution properties of conditional predictive scores, e.g. the posterior expectation \(\mathbb{E}_{\lambda|\text{data}}[S(F_\lambda, y)]\) for \(S(F_\lambda, y)\) under the posterior distribution for \(\lambda\) in the Poisson model.

For Squared Error, \[ \begin{aligned} \mathbb{E}_{\lambda|\text{data}}[(y - \lambda)^2] &= \mathbb{E}_{\lambda|\text{data}}[\{y - \mathbb{E}(\lambda|\text{data}) + \mathbb{E}(\lambda|\text{data}) - \lambda\}^2] \\ &= [y - \mathbb{E}(\lambda|\text{data})]^2 + \mathbb{E}_{\lambda|\text{data}}(\{\mathbb{E}(\lambda|\text{data}) - \lambda\}^2] \\ &= [y - \mathbb{E}(\lambda|\text{data})]^2 + \mathbb{V}(\lambda|\text{data}) . \end{aligned} \] It’s noteworthy that this is similar to the improper score \([y - \mathbb{E}(y|\text{data})]^2 + \mathbb{V}(y|\text{data})\), and also in this new case, one can have a model with artificially small posterior variance with a smaller expected score, making this type of construction problematic to interpret.

However, in some cases it does provide alternative approaches for how to compute the proper scores for the full posterior predictive distributions. If \(\lambda_{ij}\), \(j=1,\dots,J\) are samples from the posterior distribution, one score estimator is \[ \widehat{S}(F_i,y_i) = \left(y_i - \frac{1}{J}\sum_{j=1}^J \lambda_{ij}\right)^2 , \] with the averaging over the samples inside the quadratic expression, and we can use

    pred_pois <- predict(fit_pois, newdata_pois, formula = ~ exp(Intercept + x))
    scores <- (newdata_pois$y - pred_pois$mean)^2

If we instead take advantage of the new expression above, we have \[ [y - \mathbb{E}(\lambda|\text{data})]^2 = \mathbb{E}_{\lambda|\text{data}}[(y - \lambda)^2] - \mathbb{V}(\lambda|\text{data}) \] so the score can be estimated by

    pred_pois <- predict(fit_pois, newdata_pois,
      formula = ~ {
        lambda <- exp(Intercept + x)
        list(
          cond_scores = (y - lambda)^2,
          lambda = lambda
        )
      },
      n.samples = 2000
    )
    scores <- pred_pois$cond_scores$mean - pred_pois$lambda$sd^2
    summary(scores)
    #>      Min.   1st Qu.    Median      Mean   3rd Qu.      Max.
    #>  -0.00021   0.56927   3.22481  18.15547  11.34509 557.53135

For this particular case, this approach is unlikely to be an improvement or more accurate than the basic estimator. Remarkably, this estimator of Squared Error isn’t even guaranteed to be positive! However, for other scores there may potentially be practical benefits.

### An alternative estimator for CRPS

For the CRPS score, there are closed form expressions available for some distributions, conditionally on their parameters, but not for the full predictive mixture distribution. We take a similar approach as for SE, and let \(F\) and \(F_\lambda\) denote the unconditional and conditional cumulative distribution functions for the posterior predictive distribution. Then \(F(x)=\mathbb{E}_{\lambda|\text{data}}[F_\lambda(x)]\) for all \(x\), and \[ \begin{aligned} S_\text{CRPS}(F,y) &= \int_{-\infty}^\infty [F(x) - I(y\leq x)]^2 \,\mathrm{d}x \\ &= \mathbb{E}_{\lambda|\text{data}}\left[ \int_{-\infty}^\infty [F_\lambda(x) - I(y\leq x)]^2 \,\mathrm{d}x \right] - \int_{-\infty}^\infty \left\{\mathbb{E}_{\lambda|\text{data}}\left[F_\lambda(x)^2\right] - \mathbb{E}_{\lambda|\text{data}}[F_\lambda(x)]^2\right\} \,\mathrm{d}x \\ &= \mathbb{E}_{\lambda|\text{data}}\left[ S_\text{CRPS}(F_\lambda,y) \right] - \int_{-\infty}^\infty \mathbb{V}_{\lambda|\text{data}}[F_\lambda(x)] \,\mathrm{d}x . \end{aligned} \] Note that we didn’t need to use any particular model properties here, so this holds for any predictive model with mixture structure, when \(\lambda\) is the collection of model parameters. We also note the resemblance to the alternative expression for the Squared Error; this is because CRPS can be seen as the integral over all Brier scores for predicting event indicators of the form \(z=I(y\leq x)\), with probability \(F(x)\).

In the Poisson case, we can now estimate the CRPS scores like this, that makes the code a bit easier than the previous version that needed `generate()`. We can use the Poisson CRPS implementation `crps_pois()` from the `scoringRules` package.

However, it can be shown that the two approaches have nearly identical Monte Carlo variance, so the previous version is likely preferable as it doesn’t require knowing a closed form CRPS expression.

    max_K <- 100 # some large value, so that 1-F(K) is small
    pred_pois <- predict(fit_pois, newdata_pois,
      formula = ~ {
        lambda <- exp(Intercept + x)
        list(
          crps = scoringRules::crps_pois(y, lambda),
          F = do.call(
            c,
            lapply(
              seq_along(y),
              function(i) {
                ppois(seq(0, max_K), lambda = lambda[i])
              }
            )
          )
        )
      },
      n.samples = 2000
    )
    crps_score <-
      pred_pois$crps$mean -
      (pred_pois$F |>
        mutate(i = rep(seq_along(newdata_pois$y), each = max_K + 1)) |>
        group_by(i) |>
        summarise(F_var = sum(sd^2), .groups = "drop") |>
        pull(F_var))
    summary(crps_score)
    #>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
    #>  0.1965  0.7624  1.1869  1.9697  2.0782 18.1693

Formulas and functions for Poisson CRPS, as well as for other distributions, can be found via the “Evaluating Probabilistic Forecasts with scoringRules” vignette of the `scoringRules` package, and the references in `?scoringRules::crps.numeric`.